The difference with the synthesis formulation is that the variables are now \(\mathsf {s}\)  and \(\mathsf {g_H}\) , that is, galaxy images in direct space. First, the number of effective free parameters (before regularisation is applied) is largely reduced when solving in direct space: it is simply the number of pixels \(N^2\)  as opposed to \(J\times N^2\)  for \(J\)  decomposition scales. In this regard, optimisation in the analysis formulation is simpler and more stable. Second, the positivity constraint is simple to apply in direct space, because it does not require sub-iterations like in transformed space. Third, some specific features, such as a those similar to a dirac function, are better represented in direct space, whereas they may require many more coefficients in transformed space. One drawback of the analysis formulation is that the sparsity constraint is not mathematically described by the exact soft-thresholding operation, compared to the synthesis formulation. In practice, however, the process of sequentially applying the starlet transform, the soft-thresholding operation, and then the inverse starlet transform does not prevent the algorithm from converging. There exist algorithms capable of solving Eq. REF  without this approximation [1]}, but their implementation is left for a future version.
